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waves_01
WAVES
What is a wave?
A disturbance that propagates
 energy transferred
Examples
• Waves on the surface of water
• Sound waves in air
• Electromagnetic waves
• Seismic waves through the earth
• Electromagnetic waves can propagate through a vacuum
• All other waves propagate through a material medium
(mechanical waves). It is the disturbance that propagates - not
the medium - e.g. water waves
CP 478
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waves_01: MINDMAP SUMMARY
Wave, wave function, harmonic, sinusodial functions (sin, cos), harmonic
waves, amplitude, frequency, angular frequency, period, wavelength,
propagation constant (wave number), phase, phase angle, radian, wave speed,
phase velocity, intensity, inverse square law, transverse wave, longitudinal
(compressional) wave, particle displacement, particle velocity, particle
acceleration, mechanical waves, sound, ultrasound, transverse waves on
strings, electromagnetic waves, water waves, earthquake waves, tsunamis
2 
 2
 2
y ( x, t )  A sin(k x   t )  A sin 
x
t   sin 
 x  v t  
T 
 
 

2
2
1
 
k

 2 f
f 
v f  

T
T
T k
y
2 y
vp 
ap  2
t
t
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SHOCK WAVES CAN SHATTER KIDNEY STONES
Extracorporeal shock wave lithotripsy
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SEISMIC WAVES (EARTHQUAKES)
S waves (shear waves) – transverse waves that travel through the body of the
Earth. However they can not pass through the liquid core of the Earth. Only
longitudinal waves can travel through a fluid – no restoring force for a transverse
wave. v ~ 5 km.s-1.
P waves (pressure waves) – longitudinal waves that travel through the body of
the Earth. v ~ 9 km.s-1.
L waves (surface waves) – travel along the Earth’s surface. The motion is
essentially elliptical (transverse + longitudinal). These waves are mainly
responsible for the damage caused by earthquakes.
Tsunami
If an earthquakes occurs under the ocean it can produce a tsunami (tidal wave).
Sea bottom shifts  ocean water displaced  water waves spreading out from
disturbance very rapidly v ~ 500 km.h-1,  ~ (100 to 600) km, height of wave ~ 1m
 waves slow down as depth of water decreases near coastal regions  waves
pile up  gigantic breaking waves ~30+ m in height.
1883 Kratatoa - explosion devastated coast of Java and Sumatra
v  gh
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11:59 am Dec, 26 2005: “The moment that changed the world:
Following a 9.0 magnitude earthquake off the coast of Sumatra, a massive
tsunami and tremors struck Indonesia and southern Thailand Lanka - killing
over 104 000 people in Indonesia and over 5 000 in Thailand.
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Waveforms
Wavepulse
An isolated disturbance
Wavetrain
e.g. musical note of
short duration
Harmonic wave: a sinusoidal disturbance of constant
amplitude and long duration
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Wavefronts
• A wavefront is a line or surface that joins points of same
phase
• For water waves travelling from a point source, wavefronts
are circles (e.g. a line following the same maximum)
• For sound waves emanating from a point source the wave
fronts are spherical surfaces
wavefront
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A progressive or traveling wave is a self-sustaining disturbance of a medium that
propagates from one region to another, carrying energy and momentum. The
disturbance advances, but not the medium.
The period T (s) of the wave is the time it takes for one wavelength of the wave to
pass a point in space or the time for one cycle to occur.
The frequency f (Hz) is the number of wavelengths that pass a point in space in
one second.
The wavelength  (m) is the distance in space between two nearest points that are
oscillating in phase (in step) or the spatial distance over which the wave makes one
complete oscillation.
The wave speed v (m.s-1) is the speed at which the wave advances
v = x / t =  / T =  f
Amplitude (A or ymax) is the maximum value of the disturbance from equilibrium
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Harmonic wave - period
• At any position, the disturbance is a sinusoidal function of
time
displacement
• The time corresponding to one cycle is called the period T
T
A amplitude
time
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Harmonic wave - wavelength
• At any instant of time, the disturbance is a sinusoidal
function of distance
displacement
• The distance corresponding to one cycle is called the
wavelength 

A amplitude
distance
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Wave velocity - phase velocity
x 
v
  f
t T
t 0
tT

t  2T

t  3T

0  2 3

Propagation velocity (phase velocity)
distance
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Wave function (disturbance)
e.g. for displacement y is a function of distance and time
 2

y ( x, t )  A sin  ( x  v t ) 


  x t 
 A sin  2    
   T 
 A sin(k x   t )
+ wave travelling to the left
-
wave travelling to the right
Note: could use cos instead of sin
CP 484
SINUSOIDAL FUNCTION – harmonic waves
Change in amplitude A
A = 0 to 10
x=0
=0
T=2
t = 0 to 8
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angle in radians
2
 2

y  A sin 
t
x  

 T

angle in radians
SINUSOIDAL FUNCTION
Change in period T
f 
1
T
T 
A = 10
x=0
=0
T = 1 to 4
t = 0 to 8
f 
T 
f 
2
 2

y  A sin 
t
x  

 T

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SINUSOIDAL FUNCTION
Change in initial phase 
A = 10
x=0
 = 0 to 4
T=2
t = 0 to 8
angle in radians
2
 2

y  A sin 
t
x  

 T

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Sinusoidal travelling wave moving to the right
  x t 
 2

y ( x, t )  A sin  ( x  v t )   A sin 2      A sin(k x   t )


   T 
Each particle
does not move
forward, but
oscillates,
executing SHM.
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Amplitude A of the disturbance (max value measured from
equilibrium position y = 0). The amplitude is always taken as a
positive number. The energy associated with a wave is
proportional to the square of wave’s amplitude. The intensity I of
a wave is defined as the average power divided by the
perpendicular area which it is transported. I = Pavg / Area
angular wave number (wave number) or propagation constant
or spatial frequency,) k [rad.m-1]
angular frequency  [rad.s-1]
Phase (k x ±  t) [rad]
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CP 484
INTENSITY I [W.m-2]
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• Energy propagates with a wave - examples?
• If sound radiates from a source the power per unit area (called
intensity) will decrease
• For example if the sound radiates uniformly in all directions, the
intensity decreases as the inverse square of the distance from
the source.
P
P
1
I 
 2
2
A 4 r
r
Inverse square law
Wave energy: ultrasound for blasting gall stones, warming tissue in physiotheraphy; sound
of volcano eruptions travels long distances
CP 491
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The faintest sounds the human ear can detect at a frequency of
1 kHz have an intensity of about 1x10-12 W.m-2 – Threshold of hearing
The loudest sounds the human ear can tolerate have an intensity
of about 1 W.m-2 – Threshold of pain
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Longitudinal & transverse waves
Longitudinal (compressional) waves
Displacement is parallel to the direction of propagation
waves in a slinky; sound; earthquake waves P
Transverse waves
Displacement is perpendicular to the direction of propagation
electromagnetic waves; earthquake waves S
Water waves
Combination of longitudinal & transverse
 2

y ( x, t )  A sin  ( x  v t )   A sin  2 ( x /   t / T )   A sin(k x   t )


Wavelength

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[m]
y(0,0) = y(,0) = A sin(k ) = 0
k = 2
 = 2 / k
Period T [s]
y(0,0) = y(0,T) = A sin(- T) = 0
 T = 2 T = 2 / 
f = 2 / 
Phase speed v [m.s-1]
v = x / t =  / T =  f =  / k
CP 484
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As the wave travels it retains its shape and therefore, its value of the
wave function does not change i.e. (k x -  t) = constant  t
increases then x increases, hence wave must travel to the right (in
direction of increasing x). Differentiating w.r.t time t
k dx/dt -  = 0 dx/dt = v =  / k
As the wave travels it retains its shape and therefore, its value of the
wave function does not change i.e. (k x +  t) = constant  t
increases then x decreases, hence wave must travel to the left (in
direction of decreasing x). Differentiating w.r.t time t
k dx/dt +  = 0 dx/dt = v = -  / k
CP 492
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Each “particle / point” of the wave oscillates with SHM
particle displacement: y(x,t) = A sin(k x -  t)
particle velocity:
y(x,t)/t = - A cos(k x -  t)
velocity amplitude:
vmax =  A
particle acceleration: a = ²y(x,t)/t²
= -² A sin(k x -  t)
= -² y(x,t)
acceleration amplitude: amax = ² A
CP 492
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Transverse waves - electromagnetic, waves on strings, seismic - vibration at
right angles to direction of propagation of energy
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t=T
16
14
12
10
8
6
4
t
2
t=0
0
-2
0
10
20
30
40
x
50
60
70
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Longitudinal (compressional) - sound, seismic - vibrations along or parallel to
the direction of propagation. The wave is characterised by a series of alternate
condensations (compressions) and rarefractions (expansion
t = T 16
14
12
10
8
6
4
t
2
t=0
0
0
10
20
30
40
x
50
60
70
80
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34567
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Problem solving strategy: I S E E
Identity:
What is the question asking (target variables) ?
What type of problem, relevant concepts, approach ?
Set up:
Diagrams
Equations
Data (units)
Physical principals
PRACTICE ONLY
MAKES PERMANENT
Execute: Answer question
Rearrange equations then substitute numbers
Evaluate:
Check your answer – look at limiting cases
sensible ?
units ?
significant figures ?
Problem 1
For a sound wave of frequency 440 Hz, what is the
wavelength ?
(a) in air (propagation speed, v = 3.3×02 m.s-1)
(b) in water (propagation speed, v = 1.5×103 m.s-1)
[Ans: 0.75 m, 3.4 m]
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Problem 2 (PHYS 1002, Q11(a) 2004 exam)
A wave travelling in the +x direction is described by the
equation
y  0.1sin 10 x  100 t 
where x and y are in metres and t is in seconds.
Calculate
(i)
(ii)
(iii)
(iv)
the wavelength,
the period,
the wave velocity, and
the amplitude of the wave
[Ans: 0.63 m, 0.063 s, 10 m.s-1, 0.1 m]
use the ISEE method
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Problem 3
A travelling wave is described by the equation
y(x,t) = (0.003) cos( 20 x + 200 t )
where y and x are measured in metres and t in seconds
What is the direction in which the wave is travelling?
Calculate the following physical quantities:
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angular wave number
2
wavelength
use the ISEE method
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angular frequency
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frequency
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period
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wave speed
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amplitude
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particle velocity when x = 0.3 m and t = 0.02 s
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particle acceleration when x = 0.3 m and t = 0.02 s
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Solution I S E E
y(x,t) = (0.003) cos(20x + 200t)
The general equation for a wave travelling to the left is y(x,t) = A.sin(kx + t + )
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2
3
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k = 20 m-1
 = 2 / k = 2 / 20 = 0.31 m
 = 200 rad.s-1
=2f
f =  / 2 = 200 / 2 = 32 Hz
T = 1 / f = 1 / 32 = 0.031 s
v =  f = (0.31)(32) m.s-1 = 10 m.s-1
amplitude A = 0.003 m
x = 0.3 m t = 0.02 s
8 vp = y/t = -(0.003)(200) sin(20x + 200t) = -0.6 sin(10) m.s-1 = + 0.33 m.s-1
9 ap = vp/t = -(0.6)(200) cos(20x + 200t) = -120 cos(10) m.s-2 = +101 m.s-2
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